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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a></span><br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex4.html#D23" target="_self" title="CONVEX4:def.23">CONVEX4:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) &amp; <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:87"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span></span><br/>

<a NAME="E3:87"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="clvect_1.html#D6" title="CLVECT_1:def.6">CLVECT_1:def 6</a></i>;<br/>
<a NAME="E4:87"/><b>then </b><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:87"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="clvect_1.html#D6" title="CLVECT_1:def.6">CLVECT_1:def 6</a></i>;<br/>
<a NAME="E6:87"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:87"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="clvect_1.html#T2" title="CLVECT_1:th.2">CLVECT_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#T2" title="CLVECT_1:th.2">CLVECT_1:2</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E8:87"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:87"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K17" title="CONVEX4:func.17">Up</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K3" title="CLVECT_1:func.3">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D6" title="CLVECT_1:def.6">CLVECT_1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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